scholarly journals Traveling Wave Solutions of Two Nonlinear Wave Equations by (G′/G)-Expansion Method

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Yazhou Shi ◽  
Xiangpeng Li ◽  
Ben-gong Zhang
Keyword(s):  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Wave Equations ◽  
Hyperbolic Function ◽  
Exact Traveling Wave Solutions ◽  
Function Solution ◽  
Wave Solutions

We employ the (G′/G)-expansion method to seek exact traveling wave solutions of two nonlinear wave equations—Padé-II equation and Drinfel’d-Sokolov-Wilson (DSW) equation. As a result, hyperbolic function solution, trigonometric function solution, and rational solution with general parameters are obtained. The interesting thing is that the exact solitary wave solutions and new exact traveling wave solutions can be obtained when the special values of the parameters are taken. Comparing with other methods, the method used in this paper is very direct. The (G′/G)-expansion method presents wide applicability for handling nonlinear wave equations.

EXACT SOLUTIONS AND THEIR DYNAMICS OF TRAVELING WAVES IN THREE TYPICAL NONLINEAR WAVE EQUATIONS

2009 ◽  
Vol 19 (07) ◽  
pp. 2249-2266 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG ◽  
GUANRONG CHEN
Keyword(s):  
Nonlinear Wave ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Wave Equations ◽  
Visual Illusion ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Dynamical Analysis ◽  
Nonlinear Wave Equations ◽  
Wave Solutions

It was reported in the literature that some nonlinear wave equations have the so-called loop- and inverted-loop-soliton solutions, as well as the so-called loop-periodic solutions. Are these true mathematical solutions or just numerical artifacts? To answer the question, this article investigates all traveling wave solutions in the parameter space for three typical nonlinear wave equations from a theoretical viewpoint of dynamical systems. Dynamical analysis shows that all these loop- and inverted-loop-solutions are merely visual illusion of numerical artifacts. To reveal the nature of such special phenomena, this article also offers the mathematical parametric representations of these traveling wave solutions precisely in analytic forms.

scholarly journals New Exact Traveling Wave Solutions of the Time Fractional Complex Ginzburg-Landau Equation via the Conformable Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Zhao Li ◽  
Tianyong Han
Keyword(s):  
Traveling Wave ◽  
Landau Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Wave Transformation ◽  
Hyperbolic Function ◽  
Ginzburg Landau ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Ginzburg Landau Equation

In this study, the exact traveling wave solutions of the time fractional complex Ginzburg-Landau equation with the Kerr law and dual-power law nonlinearity are studied. The nonlinear fractional partial differential equations are converted to a nonlinear ordinary differential equation via a traveling wave transformation in the sense of conformable fractional derivatives. A range of solutions, which include hyperbolic function solutions, trigonometric function solutions, and rational function solutions, is derived by utilizing the new extended G ′ / G -expansion method. By selecting appropriate parameters of the solutions, numerical simulations are presented to explain further the propagation of optical pulses in optic fibers.

BREAKING WAVE SOLUTIONS TO THE SECOND CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS

2009 ◽  
Vol 19 (04) ◽  
pp. 1289-1306 ◽  
Author(s):  
JIBIN LI ◽  
XIAOHUA ZHAO ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Nonlinear Wave Equations ◽  
Breaking Wave ◽  
Dynamical Behaviors ◽  
Wave Solutions

The existence of breaking wave solutions of the second class of singular nonlinear wave equations is proved by methods from the dynamical systems theory. For the second class of singular nonlinear traveling wave equations, dynamical behaviors of the traveling wave solutions are completely classified and thoroughly discussed. Corresponding to some bounded orbits of the traveling systems, exact parametric representations of traveling wave solutions are derived within different parameter regions of the parameter space.

scholarly journals Exploration on traveling wave solutions to the 3rd-order klein–fock-gordon equation (KFGE) in mathematical physics

2020 ◽  
Vol 8 (1) ◽  
pp. 14 ◽  
Author(s):  
Nur Hasan Mahmud Shahen ◽  
Foyjonnesa . ◽  
Md. Habibul Bashar
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Dynamic Models ◽  
Gordon Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Analytic Solutions ◽  
Hyperbolic Function ◽  
Wave Solutions

In this paper, the -expansion method has been applied to find the new exact traveling wave solutions of the nonlinear evaluation equations (NLEEs) by utilizing 3rd-order Klein–Gordon Equation (KFGE). With the collaboration of symbolic commercial software maple, the competence of this method for inventing these exact solutions has been more exhibited. As an upshot, some new exact solutions are obtained and signified by hyperbolic function solutions, different combinations of trigonometric function solutions, and exponential function solutions. Moreover, the -expansion method is a more efficient method for exploring essential nonlinear waves that enrich a variety of dynamic models that arises in nonlinear fields. All sketching is given out to show the properties of the innovative explicit analytic solutions. Our proposed method is directed, succinct, and reasonably good for the various nonlinear evaluation equations (NLEEs) related treatment and mathematical physics also. 

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